PRAMANA
— journal of
physics
c Indian Academy of Sciences
Vol. 86, No. 2
February 2016
pp. 207–221
New minimal SO(10) GUT: A theory for all epochs
CHARANJIT S AULAKH1,2
1 Department
of Physics, Panjab University, Chandigarh 160 014, India
of Physical Sciences, IISER Mohali, Punjab 140 306, India
E-mail: [email protected]
2 Department
DOI: 10.1007/s12043-015-1141-2; ePublication: 7 January 2016
Abstract. The supersymmetric SO(10) theory (NMSO(10)GUT) based on the 210 + 126 + 126
Higgs system proposed in 1982 has evolved into a realistic theory capable of fitting the known low
energy particle physics data besides providing a dark matter candidate and embedding inflationary cosmology. It dynamically resolves longstanding issues such as fast dimension five-operator
mediated proton decay in SUSY GUTs by allowing explicit and complete calculation of crucial
threshold effects at MSUSY and MGUT in terms of fundamental parameters. This shows that SO(10)
Yukawas responsible for observed fermion masses as well as operator dimension-five-mediated proton decay can be highly suppressed on a ‘Higgs dissolution edge’ in the parameter space of GUTs
with rich superheavy spectra. This novel and generically relevant result highlights the need for
every realistic UV completion model with a large/infinite number of heavy fields coupled to the
light Higgs doublets to explicitly account for the large wave function renormalization effects on
emergent light Higgs fields. The NMSGUT predicts large-soft SUSY breaking trilinear couplings
and distinctive sparticle spectra. Measurable or near measurable level of tensor perturbations – and
thus large inflaton mass scale – may be accommodated within the NMSGUT by supersymmetric
see-saw inflation based on an LHN flat direction inflaton if the Higgs component contains contributions from heavy Higgs components. Successful NMSGUT fits suggest a renormalizable Yukawon
ultraminimal gauged theory of flavour based upon the NMSGUT Higgs structure.
Keywords. Supersymmetry; grand unification; SO(10); flavour unification; inflation.
PACS Nos 12.60.Jv; 12.10.Dm; 98.80.Cq; 11.30.Hv
1. Introduction
Grand unification theories (GUTs) have seen around 40 summers since the basic idea was
first proposed by Pati and Salam in 1974 [1], yet retain their attraction as the most obvious
progression of the gauge logic embodied in the Standard Model. Indeed, in particle
physics, the only tangible and clear hints of physics beyond the Standard Model are the
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Charanjit S Aulakh
remarkable convergence of gauge couplings in the unification regime: 1015 –1017 GeV
and the existence of neutrino masses (initially thought to be superfluous in both the
electroweak and GUT contexts). To this, one should perhaps append the convergence
of the pivotal [2] third-generation Yukawa couplings in the MSSM at high tan β. The
same gauge logic that structures the SM (and justifies the zeroth-order decoupling of
right-handed neutrinos) also makes the Type-I see-saw mechanism [3], the most natural
rationale for the milli-eV range neutrino masses actually observed. Supersymmetry seems
essential to the structural stability of the SM and experience [4,5] with supersymmetric
LR models and renormalizable SO(10) GUTs shows it ensures complete calculability for
UV completions of the MSSM. Thus, non-discovery of sparticles in the searches so far has
still not dimmed the ardor of its many adherents. Arguments for considering some variety of the MSSM as the effective theory of any GUT became all the more cogent when,
with the observation of solar and atmospheric neutrino oscillations at Super-Kamiokande,
the scale of B-L breaking associated with a renormalizable see-saw mechanism emerged
2
/(10−12 GeV) ∼ 1016 GeV) in the GUT ball-park. The convergence of these two
(VEW
compelling arguments thus hints at a deep interconnection between B and L violation in a
supersymmetric grand unified framework [6,7]. It is SO(10) not SU (5) GUTs that provide
the most natural GUT framework, free of gauge singlets (which are really opaque, and
thus anathemic, to the gauge logic pursued so successfully in the SM), for Type-I (and also
Type-II) see-saw. Thus, eventually SO(10) GUTs displaced SU (5) theories and achieved
a long-delayed vindication and recognition as minimal UV complete frameworks for neutrino oscillations. It was only natural that our pursuit of consistent minimal left–right
supersymmetric R-parity preserving models [4] from the midnineties led in short order
[5], after the epochal measurements of Super-Kamiokande to a realization that the very
first complete supersymmetric SO(10) GUT [8,9] proposed way back in 1982, just after
the Georgi–Dimopoulos minimal SUSY SU(5) model [10], was in fact the minimal supersymmetric GUT (MSGUT) [11]. This model is now called [12] – due to a transient
glory of a truncated version with only 10 + 126 but not 120 coupling to matter fermion
bilinears as the minimal theory [13] – the New/Next MSGUT(NMSGUT). In the best
Popperian mode it has survived another decade of detailed investigation of its ability
to fit all available SM, gauge, fermion and neutrino data, as well as the consistency of
one-loop threshold corrections at both low- and high-energy scales made possible by its
extreme parameter economy (a.k.a. minimality) and full calculability: simple virtues that
are, alas, all too rare among the plethora of its (non-minimal) competitors. The model is
based on a 210 ⊕ 126 ⊕ 126 GUT Higgs system and already in 1982 [8] showed clearly
that prevention of a RG flow catastrophe due to large pseudo-Goldstone supermultiplets
requires a single step breaking of SUSY SO(10) to the MSSM: later rediscovered in the
context of another R-parity preserving but non-minimal model [5]. Being a complete theory of gauge physics, the model is capable of fitting and/or predicting most, if not all, of
the commonly considered varieties of BSM physics, including neutrino masses, the g-2
muon anomaly, LSP dark matter, B violation (at acceptable rates: see below), leptogenesis
driven by the parameters controlling neutrino mass etc. Remarkably, it also comes close
to providing a workable embedding of inflation based on the D-flat directions involving
lepton and Higgs doublet fields naturally present and contributing to the effective MSSM
below the GUT scale. The model also predicts a distinctive and characteristic normal
hierarchy of sfermion generations discoverable by the LHC or its successors. As striking
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New minimal SO(10) GUT
was its early indication, on the basis of the parameters required for a successful fermion
fit, that the trilinear soft SUSY breaking parameter A0 must be large (as far back as 2008
[14], i.e. well before Higgs mass discovery made such a large A0 respectable rather than
ludicrously fine tuned: as then argued by naturalists).
The Standard Model parameters are fitted numerically in terms of definite sets of grand
unified model coupling. For each set of couplings adequate for a fit within the experimental and theoretical accuracy, the model makes predictions for a vast range of phenomena
ranging from exotic processes such as baryon number and lepton flavour violation to
cosmological leptogenesis, dark matter relic density and inflation. The demonstration
of the feasibility of such comprehensive fits of the SM data directly in terms of SUSY
GUT parameters marks the NMSGUT as being cast in the mould of the SM rather than
its piece-meal generalizations. The most striking themes and results that have emerged
recently from our detailed investigation of the structural freedoms-in-necessity granted by
the full calculability NMSGUT are:
(1) A simple generic mechanism for suppression of the long problematic [15] d = 5
fast proton decay generic to SUSY GUTs.
(2) An embedding of high-scale supersymmetric renormalizable inflexion [16] based
on the left lepton (L)–Higgs doublet (H)–conjugate neutrino (N ≡ νLc ) flat direction (labelled by the LHN chiral invariant) and utilizing the involvement of heavy
partners of the MSSM Higgs. This may prove very attractive in the context of the
BICEP2-driven focus on high scale inflation.
(3) Completely novel ‘grand Yukawonification’ models based on gauging of an O(3)
subgroup of the U (3) flavour symmetry of the SO(10) GUT fermion kinetic terms.
This flavour symmetry is naturally broken at the GUT scale by the NMSGUT
Higgs fields which promote themselves rather naturally to also carry the function
of being Yukawons i.e., fields whose VEVs determine the Yukawa couplings in
the effective MSSM besides simultaneously breaking SO(10) while maintaining
renormalizability: which is not possible in most Yukawon models.
(4) The models in point (3) above are made possible by a novel conflation of
supergravity-mediated supersymmetry breaking with the breaking of gauged flavour symmetry in the hidden sector of the model using the so-called Bajc–Melfo
calculable metastable SUSY breaking vacua [17,18].
Thus, the NMSGUT potentially provides a completely realistic and predictive theory
of particle physics in all energy ranges and cosmological epochs. We have even speculated [19] that the Landau pole in the NMSGUT gauge coupling which occurs quite near
the Planck scale should be interpreted as a physical cut-off on the perturbative dynamics
associated with the scale where the NMSGUT condenses (via a strong coupled supersymmetric dynamics). This scale could then function as the Planck scale of an effective
induced supergravity emergent at large length scales from the NMSGUT when the metric and gravitino fields introduced to define a coordinate independent microscopic GUT
acquire kinetic terms due to quantum effects. Since points (2) and (3) are published as
[16,17] and are also reported in these proceedings in the contributions of Ila Garg and
Charanjit Kaur respectively, I shall touch upon them only briefly but focus on the results
regarding points (1) and (4) and discuss some of their implications while referring the
reader to the published papers [11,12,20–23] for details.
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Charanjit S Aulakh
2. NMSGUT basics
The NMSGUT superpotential is built from the quadratic SO(10) invariants and associated
mass parameters
m : 2102 ;
M : 126 · 126;
MH : 102 ;
m : 1202
(1)
and trilinear couplings
λ : 2103 ; η : 210 · 126 · 126; ρ : 120 · 120 · 210
k : 10 · 120 · 210; γ ⊕ γ̄ : 10 · 210 · (126 ⊕ 126)
ζ ⊕ ζ̄ : 120 · 210 · (126 ⊕ 126)
16A .16B .(hAB 10 + fAB 126 + gAB 120).
(2)
The couplings h, f (g) are complex (anti)-symmetric in the flavour indices due to properties of the SO(10) Clifford algebra. Either h or f is chosen as real and diagonal using the
U (3) flavour symmetry of the matter kinetic terms. The matter Yukawas thus contain 21
real parameters. Five phases say of m, M, λ, γ , γ̄ are set to zero by phase conventions.
One complex parameter (say MH ), is fine-tuned to keep two Higgs doublets (H, H̄ ) of the
effective MSSM light, leaving 23 magnitudes and 15 phases as parameters. Fine tuning
fixes H, H̄ composition as a mixture of (six pairs of the MSSM type) doublet fields in the
GUT as a function of the superpotential parameters entering the null vectors of the G3,2,1
irreps [1, 2, ±1] mass terms. The mixture is described by the so-called ‘Higgs fractions’
[11,21].
The GUT scale VEVs (units m/λ) are known functions of x which is a solution of the
cubic (ξ = λM/ηm)
8x 3 − 15x 2 + 14x − 3 + ξ(1 − x)2 = 0.
(3)
The complete set of GUT scale mass matrices for the 26 different MSSM irrep types
and tree level low-energy effective superpotential was calculated [12] extending the result
for the MSGUT [20,21,23,24]. Using these and two-loop RG flows the gauge, superoptential and soft SUSY breaking parameters at GUT scales can be matched, using a downhill
simplex search procedure, to the known values of the MSSM data described above [12].
This also yields a mini-split (10–100 TeV) supersymmetry sparticle spectrum with large
A0 , μ parameters, light gauginos and Bino LSP, superheavy Higgsinos, sfermions in tens
of TeV, normal inter-generational hierarchy and sometimes an smuon (or other sfermion)
light enough (i.e., within 10% of the light Bino LSP) to co-annihilate with it and provide
acceptable dark matter relic density [25]. The light smuon case is obviously attractive if
supersymmetry is called upon to explain the muon magnetic moment anomaly and emerges
in the right ball park for these solutions. Flavour violation in the quark and lepton sectors
is also well controlled because of the multi-TeV masses of most of the sfermions. Moreover, A0 is required to be large to allow the b quark mass to be fitted and this was found
[12,14] serendipitously before Higgs discovery made it a requirement for supersymmetry.
3. GUT scale threshold corrections and baryon decay rate
The SO(10) parameters determined by the realistic fit are substituted into the effective
(dimension four) superpotential describing baryon violation which, after RG flow to low
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New minimal SO(10) GUT
energies, is used to calculate the proton decay rate. As is well known, the generic result
gives a lifetime of some 1027 years i.e., some seven orders of magnitude shorter than
the current limits from the Super-Kamiokande experiment. The beautifully complete and
predictive fits are thus of little use if this problem remains unsurmountable. However, as in
previous tight spots, the NMSGUT points out a convincing and illuminating, generically
applicable and dynamical, pathway out of the difficulty: precisely because of the available
explicit solution described above.
Superpotential parameters renormalize only by wave function corrections. In the computable basis sets where heavy supermultiplet masses are diagonal, a generic heavy field
¯ mass matrix diagonalizes as
type (conjugate irrep )
¯ = U ¯ ′ ;
= V ′
⇒
¯ T M = ¯ ′ T MDiag ′ .
(4)
Circulation of heavy fields within the quantum loops which normalize the propagators of the light fields (fA , fAc , Hf ) which enter Higgs matter Yukawa couplings
in the Lagrangian: L = [fcT Yf f Hf ]F + h.c.+ implies [26] a finite wave function
renormalization in the fermion and Higgs kinetic terms
†
†
B
B
†
†
(f¯A (Zf¯ )A f¯B + fA (Zf )A fB ) + H ZH H + H̄ ZH̄ H̄ +· · · . (5)
L=
A,B
D
Unitary matrices UZf , ŪZf¯ diagonalize (U † ZU = Z ) Zf,f¯ to the positive definite
form Zf ,Zf¯ . We define a new basis to put the kinetic terms of the light matter and Higgs
fields in canonical form:
−1/2
f¯ = UZf¯ Zf¯ f˜¯ = ŨZf¯ f˜¯
−1/2
f = UZf Zf f˜ = ŨZf f˜;
H̃
H = √ ;
ZH
H̄˜
H̄ = .
ZH̄
(6)
Thus when matching the MSSM to the effective theory derived from the NMSGUT it is
the loop corrected Yukawa couplings Ỹf of the new (canonically normalized) basis fields
f˜, f,˜¯ H̃ , H̄˜ , and not the original ones (Y ) derived from the GUT at tree level, that must
f
be equated to the MSSM couplings at the matching scale
Yf
Yf
−1/2
−1/2
Ỹf = Zf¯ UZTf¯ UZf Zf = ŨZTf¯ ŨZf
ZHf
ZHf
(7)
and not the original tree-level ones that match the MSSM at the matching scale.
For a light field i the corrections have the form (Z = 1 − K):
j
Ki = −
2 g10
1 α∗ α
Q
Q
F
(m
,
m
)
+
Yikl Yj∗kl F (mk , ml ),
α
k
ik
kj
8π 2 α
32π 2 kl
(8)
where L = g10 Qαik ψi† γ μ Aμα ψk describes the generic gauge coupling and W =
1
Y , the generic Yukawa couplings, while F (m1 , m2 ) are symmetric 1-loop
6 ij k i j k
Passarino–Veltman functions.
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Crucially, the SO(10) Yukawa couplings (h, f, g)AB also enter into the coefficients
LABCD , RABCD of the d = 5 baryon decay operators in the effective superpotential
obtained by integrating out the heavy chiral supermultiplets that mediate baryon decay
[12,20,21].
Ỹf must be diagonalized to mass basis (denoted by primes), so that d = 5, B = ±1
decay operator coefficients become
L′ABCD =
′
RABCD
=
Labcd (ŨQ′ )aA (ŨQ′ )bB (ŨQ′ )cC (ŨL′ )dD ,
a,b,c,d
Rabcd (Ũē′ )aA (Ũū′ )bB (Ũū′ )cC (Ũd̄′ )dD .
(9)
a,b,c,d
We find that when we search for a fit to the MSSM data with the constraint that
′
L′ABCD , RABCD
be sufficiently suppressed (i.e., yielding proton lifetime τp > 1034 yr)
the search (programme) flows invariably towards parameter regions where ZH,H̄ ≪ 1.
So SO(10) Yukawa couplings required to match the MSSM are greatly reduced. The
′
quadratically thus suppressing them drastically.
same couplings enter L′ABCD , RABCD
This mechanism is generically available to realistic multi-Higgs theories. Indeed we go
so far as to say that any UV completion incapable of performing this computation will
remain less than quantitative and thus is not a viable completion proposal at all. A tedious
calculation determines the threshold corrections (see [27,28] for details). Z ≃ 0 also
leads to smaller GUT couplings and compressed heavy spectra compared to previous fits.
Higher loop corrections seem computationally prohibitive. However, we have calculated
[25,29–31] the complete SO(10) two-loop beta functions and two-loop threshold corrections also rely upon the same anomalous dimensions. So convoluting GUT scale mass
spectra with our SO(10) loop should determine two-loop threshold corrections as well.
Recovering our one-loop results by this method is the necessary first step to proceed in
this direction.
Searches for fits using the threshold-corrected baryon decay operators yield s-spectra
similar to those found earlier, but with smaller values of all couplings (because of the
constraints ZH,H̄ > 0 imposed on the searches: which are badly violated if almost any of
the superpotential parameters grow large), and acceptable d = 5 B decay rates: as shown
in table 1. Imposing Max|O (4) | < 10−22 GeV−1 gives proton lifetimes above 1034 yr.
ZH,H̄ approach zero (from above) while Zf,f¯ are close to 1: since 16-plet Yukawas are
all suppressed (see [25,27–29] for further details of these and and many related issues).
Table 1. d = 5 operator-mediated proton lifetimes τp (yr), decay rates Ŵ(yr−1 ) and
branching ratios in the dominant meson+ + ν channels.
Solution
τp (M + ν)
Ŵ(p → π + ν) BR(p → π + νe,μ,τ ) Ŵ(p → K + ν) BR(p → K + νe,μ,τ )
1
9.63 × 1034 4.32 × 10−37
2
3.52 × 1034 2.14 × 10−36
212
{1.3 × 10−3 ,
0.34, 0.66}
{1.7 × 10−3 ,
0.18, 0.81}
9.95 × 10−36
2.62 × 10−35
Pramana – J. Phys., Vol. 86, No. 2, February 2016
{4.6 × 10−4 ,
0.15, 0.85}
{1.8 × 10−3 ,
0.19, 0.81}
New minimal SO(10) GUT
Table 2. Soft SUSY breaking spectrum of the decoupled and mini-split-SUSY type
that arises in a typical realistic fit. Note the large values of μ, B, A0 . Gauginos are the
lightest sparticles, and the light Higgs is near the discovered value, but Higgsinos are
superheavy. Note specially the distinctive normal super-hierarchy and the very light
smuon.
Field
Mass (GeV)
MG̃
Mχ ±
Mχ 0
Mν̃
Mẽ
Mũ
Md̃
MA
MH ±
MH 0
Mh0
1000.14
569.81, 125591.22
210.10LSP , 569.81, 125591.20 , 125591.20
15308.069, 15258.322, 21320.059
1761.89, 15308.29, 211.57smuon , 15258.60, 20674.72, 21419.56
11271.80, 14446.76, 11270.63 , 14445.80, 24607.51, 40275.87
8402.99, 11272.10, 8401.48 , 11270.95, 40269.19, 51845.93
377025.29
377025.30
377025.28
124.00
Our fits are associated with very distinctive sparticle spectra. An example is shown in
table 2 which is of the general type used to evaluate the B-violation rates seen in table 1.
Note the peculiar and remarkable smallness of the smuon mass which is a consequence
of an RG flow driven by special features of the two-loop RG flow of soft SUSY breaking
parameters in models with MH2 > MH̄2 < 0. As discussed in [27], the RG flow in such
theories can be such as to drive Mτ̃2R first negative and then to large positive values (due to
the large value of the third-generation Yukawa coupling) while in the relatively flat evolution of the first two s-generations, the R-smuon lags the R-stau and the R-selectron the
R-smuon due to the significant difference in their Yukawa couplings (figure 1). This can
4 108
MeRsq
2 108
0
2 108
4 108
0
5
10
15
Log_10(Q)
Figure 1. Running right-slepton masses squared showing development of small Rsmuon (medium-dashed blue line almost obscured by the small-dashed red line of the
R-selectron it leads) which far lags the development of R-stau (large-dashed blue line)
which first turns negative and then becomes large driven by the large third-generation
Yukawa coupling.
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Charanjit S Aulakh
result in the peculiar feature that the running mass squared of the smuon is the smallest
of all the sfermion masses and close to the LSP mass. Then the smuon (or other light
sfermion as in other cases, up or down squarks of the light generations may emerge lightest) provides the necessary coannihilation channel for achieving an acceptable bino LSP
dark matter [25]. The smuon case is doubly attractive as it will also contribute to the
muon g-2 anomaly. We note however that the sfermion spectra shown here are not yet
loop corrected and that the loop-corrected spectra in solutions which we have found so
far do not present this feature.
Besides the immediate satisfaction of finding our model to be realistic and viable, our
results imply that the criteria that must be satisfied by candidate UV completions of the
(MS)SM should be significantly modified. Candidate models with a large or infinite number of fields must not only show that their parameters can consistently be constrained to
yield some variant of the (MS)SM, but also that modification of light wave functions by
coupling to heavy fields is both calculable and sensible in the sense of the possibilities
considered and demonstrated above. In the absence of such a calculability and consistency, a UV completion model will remain at the level of a phantasmal possibility rather
than a scientific model capable of braving this falsification gauntlet to progress to the
status of a scientific theory. We have thus shown that for the NMSGUT, careful attention to the quantum communication between the low-energy effective theory and the UV
completion through the light Higgs portal yields natural and generic suppression of fast
proton decay in SUSY GUTs [28] due to the cumulative effects of the large number of
GUT fields that allow the theory to approach and live on the ‘Higgs dissolution edge’ by
renormalizing the light Higgs wave function and thus its Yukawa couplings, even when
all individual GUT couplings are well within the perturbative regime.
4. Supersymmetric see-saw inflation
The necessity of cosmological inflation and even its description in terms of a single ‘inflaton’ slowly rolling down a potential plateau before oscillating around its true minimum
to produce quanta of the low-energy theory in the post-inflationary reheating regime is
by now well accepted [32]. However, the provenance of the inflaton field and potential is
faced by an embarrassing multiplicity of candidates: many of which are compatible [32]
with even the latest data [33]. From a particle physics – rather than gravitation physics –
viewpoint, identification of an inflaton candidate from among the known particle degrees
of freedom is most appealing. The suggestion [34] to utilize one (or more) of the D-flat
directions (labelled by the chiral gauge invariants) of the MSSM Lagrangian as an inflaton candidate is thus in vogue. Such a suggestion does however face various stringent
constraints such as ensuring a suitable scale of inflation together with the tiny Yukawa
couplings for matter fermions required to ensure flatness of the potential during inflation.
As neutrino-lepton Yukawa couplings may well be small, flat directions with sneutrino
and lepton components are well suited for this role. In [35], an MSSM extended by conjugate neutrinos (N ≡ νLc ) superfields to allow (tiny) Dirac neutrino masses and an extra
gauge generator (B–L) was used as a framework for an inflaton, made up of the lepton
and Higgs doublets (the flat direction is labelled as φ ≡ LH N ). Provided a stringent
fine tuning is made between the trilinear soft SUSY breaking Aφ and the inflaton mass,
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New minimal SO(10) GUT
φ enjoys a renormalizable potential characterized by an inflection point at φ0 ∼ mφ / hν .
With mφ ∼ 100 GeV–10 TeV, hν ∼ 10−12 . Then φ0 ∼ 1011 −1015 GeV and the energy
√
1/4
density is characterized by a scale V0 ≡ ∼ mφ / hν ∼: 108 −1010 GeV. This model
is extremely fine tuned [36] as the inflaton mass Mφ is set by the SUSY breaking scale
to lie in the TeV range. Moreover, as SUSY breaking parameters are subject to radiative corrections, it seems strange to impose extreme fine tuning on the trilinear coupling
of the inflaton which arises only from the soft supersymmetry breaking parameters. It
was natural [16] for us to ask the consequences, in this context, of using see-saw [3]
masses for the light neutrinos i.e., heavy Majorana masses for the right-handed neutrinos and consequently tiny Majorana masses for the left-handed neutrinos. We showed
that the soft parameters were then irrelevant so that tuning was necessary only in the
superpotential. The inflaton mass is then set by the large right-handed neutrino masses
(mφ ∼ Mν c ∼ 106−1012 GeV). Then superpotential Yukawas hν ∼ 10−13 (M/GeV)1/2
could thus be as large as 10−6 and have mild fine tuning: rather than the radiatively
unstable and highly fine tuned [36] soft symmetry breaking parameters of the Dirac case.
The embedding of the supersymmetric see-saw inflation (SSI) model in the NMSGUT
was also attempted but could not achieve a large enough number of e-folds. However,
the advent of BICEP2 has reopened the whole question of inflaton mass which suddenly
made it plausible that it might be anywhere in the range between Mν c and MGUT ! The
ratio of tensor to scalar modes r emerges as r ∼ 2(mφ /(1014 GeV))3 but r will be measurable in the near term only if r > 10−1.5 . If the value of r is in this ball park, then the
inflaton mass would need to be much larger than even the commonly accepted upper limit
of around 1012 GeV on the right-handed neutrino masses. Interestingly, the NMSGUT
embedding studied by us earlier allows a generalization where mass of superheavy Higgs
doublets mixed into the inflaton controls its mass allowing inflaton masses right upto the
BICEP2-indicated range. This scenario allows raising the achievable number of e-folds
by around five orders of magnitude! The contribution of Ila Garg in this volume provides
further details.
5. Grand Yukawonification and the flavour-SUSY breaking link
The NMSO(10)GUT has achieved gauge and third-generation Yukawa unification consistent with B violation limits, along with excellent fits of other known fermion data. It
makes an intriguing and distinctive prediction regarding sparticle spectra, and may well
be compatible with the dominant cold dark matter plus inflation plus leptogenesis cosmological scenario, especially if the scale of inflation turns out to be near to that suggested
by BICEP2 [37]. It is only natural to ask whether such a complete model can pretend
to shed any light on the most outstanding mystery of particle physics: the flavour puzzle
a.k.a. the fermion hierarchy i.e., the origin of the peculiar large intergenerational ratios of
charged fermion masses combined with small quark but two large lepton mixing angles.
At first sight, as we impose no discrete symmetries and as Yukawa couplings are parameters dialed to fit the data, it seems that this is asking too much. However, reflection on
the way in which the SM Yukawa couplings emerge and the fact that the only convincing hint of flavour unification seems to be the convergence of third-generation Yukawa
couplings at the GUT scale suggests that perhaps the Yukawa couplings may emerge via
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Charanjit S Aulakh
spontaneous breaking of the U (3) flavour symmetry of the SO(10) invariant matter kinetic
terms. In [17] we suggested that our work [8,11,12,21,22,28] on MSGUTs naturally and
minimally identifies MSGUT Higgs multiplets as candidate Yukawon multiplets. It thus
yields a novel mechanism whereby the fermion hierarchy could emerge from a flavour
symmetric and renormalizable GUT. In our work ‘Yukawons’ also carry representations
of the gauge (SM/GUT) dynamics. In previous work, typically the dimension-1 Yukawaon Y in the Higgs vertex makes it non-renormalizable (L = f c Yf H /Y + · · · : where
the unknown high-scale Y controls Yukawa-on dynamics). Minimal SO(10) GUTs
[8,9,11,12] provide a gauged O(Ng ) family symmetry route to ‘Yukawonifcation’– with
the GUT and family symmetry breaking at the same scale – obviating the need for nonrenormalizable interactions and any extraneous scale Y . As we shall see, the consistency
conditions for the maintainability of the Higgs portal to the UV completion of the MSSM
play a central role in determining just how the peculiarly lopsided and ‘senseless’ fermion
hierarchy is produced from the flavour symmetric and grand unified UV completion. In
minimal SUSY GUTs, one eschews invocation of discrete symmetries and insists only
upon following the logic of SO(10) gauge symmetry. This insistence, combined with
careful attention to the implications of the emergence of a single light MSSM Higgs
pair from the 2Ng (Ng + 1) pairs in the O(Ng ) extended MSGUT, leads to an effectively
unique extension of the SO(10) gauge group by a O(Ng ) family gauge symmetry for the
Ng generation case and the dynamical emergence of fermion hierarchy and mixing. The
¯ also contributes to both neutrino and charged fermion masses. Gauging just an
126()
O(Ng ) subgroup of the U (Ng ) symmetry of the fermion kinetic terms seems the workable
option, in contrast to a unitary family group, because the use of complex representations
introduces anomalies and requires doubling of the Higgs structure to cancel anomalies
and to permit holomorphic invariants to be formed for the superpotential. Worse, unitary symmetry enforces vanishing of half the emergent matter Yukawa couplings. O(Ng )
family symmetry suffers from none of these defects and gauging it ensures that no Goldstone bosons arise when it is spontaneously broken. We emphasize that in contrast with
previous ‘spurion/Yukawa-on’ models (see e.g. [38]), our model is renormalizable and
GUT-based.
The GUT superpotential has exactly the same form as the MSGUT (see [12,20,21,23]
for comprehensive details):
¯ · + η · ¯ · )
WGUT = Tr(m2 + λ3 + M ¯ + MH H · H ),
+ · H · (γ + γ̄ · )
WF = A · ((hH ) + (f ) + (g))AB B .
(10)
We have shown how the 120-plet is included in WF but have studied only MSGUTs
(i.e., with 10, 126). Inclusion of the 120-plet does not affect GUT SSB. The only innovation in Higgs structure is that all the MSGUT Higgs fields now carry symmetric
¯ , H }AB ; A, B = 1, 2..Ng (under
representations of the O(Ng ) family symmetry: {, ,
which the matter 16-plets ψA are vector Ng -plets). Couplings h, f, g are single complex
¯ and antisymmetric () represennumbers, while the Yukawons carry symmetric (H, )
tations of O(3): as required by the transposition property of relevant SO(10) invariants.
For Ng = 3, real fermion mass parameters come down from 15 (Re[hAA ], fAB ) to just 3
(Re[h], f ) without the 120-plet (6 additional to just 2 with the 120-plet). Thus, this type
of renormalizable flavour-unified GUTs can legitimately be called Yukawon ultraminimal
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New minimal SO(10) GUT
GUTs (YUMGUTs). Further details on the gauge symmetry breaking and determination of Yukawas can be found in the contribution of Charanjit Kaur in this volume.
We shall focus on our resolution [18] of a severe technical difficulty that crops up when
we spontaneously break a gauged O(3) subgroup of the U (3) supersymmetric flavour
symmetry. If this difficulty is resolved, the determination of the viability of the ‘grand
Yukawonification’ model then becomes a matter of searching the relatively small remaining parameter space for viable parameter sets that fit the fermion data at MX , while taking
account of threshold corrections at low and high scales and while respecting constraints
on crucial quantities like the proton lifetime [12,21,28]. Note that in this approach, not
only are the hard parameters of the visible sector superpotential reduced by replacement
of the flavoured parameters by bland family symmetric ones but also the soft supersymmetry breaking parameters are determined by the two parameters of the hidden sector
superpotential and the Planck scale.
With just the family index carrying MSGUT Higgs present, the flavour D-terms cannot
vanish. So additional fields with VEVs free to cancel the contribution D̄XA of the GUT
sector to O(Ng ) D terms are needed. The extra F terms must be sequestered from the
GUT sector to preserve the MSGUT SSB. The special role of the Bajc–Melfo supersymmetry breaking is that it provides flat directions in both the singlet and the gauge-variant
parts of a symmetric chiral supermultiplet SAB = SBA . As it is very difficult to make
the contribution of the visible sector GUT fields to the O(Ng ) D-terms vanish, the Ŝ
flat direction performs the invaluable function of cancelling this contribution without disturbing the symmetry breaking in the visible sector. We proposed [18] Bajc–Melfo-type
two-field superpotentials [39,40] (of structure W = S(μB φ + λB φ 2 )). Their potentials
have local minima breaking supersymmetry (FS = 0) which leaves the VEV S undetermined (φ gets a VEV). If φ, S transform in symmetric representations of O(Ng ), then
the undetermined VEV of the trace-free part of S is determined at the global supersymmetry level by the minimization of the O(Ng ) D-terms thus preserving supersymmetry from
high-scale breaking. On the other hand, BM superpotentials and their metastable (local)
supersymmetry breaking vacua function efficiently [18] as hidden sectors for supergravity GUT models: coupling to supergravity determines the O(Ng ) singlet part of S to
have a VEV of order of the Planck scale. This felicitous and unexpected marriage of the
NMSGUT to gauged flavour symmetry and supergravity, reduces the number of SO(10)
Yukawas drastically and makes a quite novel linkage between spontaneous violation of
flavour and supersymmetry breaking. As the traceless parts of the fermionic components
of S are massless at tree level (the trace part is eaten by the massive gravitino) they get
masses only via radiative corrections and tend to remain very light. They could thus provide very light dark matter candidates as well as cause cosmological problems of the sort
typically associated with moduli fields [41]. This requires further detailed study but it is
interesting that such a simple extension of the NMSGUT can both reduce free parameters and provide moduli dynamics by taking on the meta-problem of the origin of flavour
hierarchy. The structure used entails yet further stringent constraints as the masslessness
of the moduli multiplets ŜAB before supersymmetry breaking implies the existence of
Ng (Ng + 1)/2 − 1 SM singlet fermions generically lighter than the gravitino mass scale
and possibly as light as a few GeV. In addition, the Polonyi mode Ss may also lead to
difficulties in the cosmological scenario. Thus, such modes can be both a boon and a
curse for familion GUT models: a boon because generic SUSY GUT models are hard
Pramana – J. Phys., Vol. 86, No. 2, February 2016
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Charanjit S Aulakh
put if asked to provide SUSY WIMPs of mass below 100 GeV as CDM candidates as
suggested by the DAMA/LIBRA experiment [42] and a curse because there are strong
constraints on the existence of such light moduli which normally demand that their mass
be rather large (>10 TeV) due to the robust cosmological (Polonyi) problems arising from
decoupled modes with Planck scale VEVs [43]. In contrast to the simple Polonyi model
and string moduli, the BM moduli have explicit couplings to light fields through family
D-term mixing and loops. Moreover, the MSGUT scenario favours [12,28] large gravitino
masses >5−50 TeV. Thus, the Polonyi and moduli problems may be evaded. In any case,
the cosmology needs to be considered seriously only after we have shown that the MSSM
fermion spectrum is indeed generated by the ‘Yukawonified’ NMSGUT [44].
6. Discussion
The vast range of applicability of the NMSGUT, justifies its claim that it is a theory for all
epochs. Not only does it present a well-controlled framework for realizing the longstanding dream of grand unification in a completely realistic fashion, it also makes distinctive
predictions concerning the observation of supersymmetry and lepton flavour violation
and dynamically palliates the proton decay problem that has chronically afflicted SUSY
GUTs without introducing extraneous structures but in a way intrinsic to the GUT itself.
Furthermore, it provides a surprisingly simple context for embedding high-scale supersymmetric inflation. The right-handed neutrinos and adequate, CKM–PMNS-linked, CP
violation that are present can enable leptogenesis. Reaching still further, it even serves as
a basis for the generation of flavour hierarchy by the spontaneous violation of a gauged
flavour symmetry at the GUT scale and in the process makes the startling suggestion
that supersymmetry breaking is linked to flavour violation via a novel BM hidden sector. This implies a striking and novel (for SUSY GUTs) prediction of possibly very
light non-neutralino dark matter candidates. Interestingly, the self-coupling of this light
DM is not constrained. The working out of the various flavour breaking scenarios while
maintaining realism in the fermion and GUT SSB sector and accounting for low- and
high-scale threshold corrections is indeed a formidable but manageable project that has
already cleared several hurdles [12,22,28,45] that might well have falsified this model
in the three decades since its proposal. These successes motivate our belief that a focus
on the implication of the consistency conditions that define our world as we know it,
in the context of the flavour symmetric NMSGUT, may further extend the scope of the
NMSGUT to include a dynamical understanding of the flavour hierarchy. Moreover, the
strong reduction in the number of parameters makes falsification much more practicable.
Although it continues to live dangerously, we hope that this will be seen as a scientific
virtue of the NMSGUT and its generalizations rather than as an unnecessary concreteness
and optimism, with which it is sometimes reproached.
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